Convolution Smoothed Quantile Regression for XGBoost

Mandy Yao; Meredith Franklin

stat.MLAug 15, 2026

Convolution Smoothed Quantile Regression for XGBoost

Authors: Mandy Yao, Meredith Franklin

Organizations: University of Toronto · Department of Statistical Sciences, University of Toronto

Abstract

The increasing availability of large and complex datasets across many scientific disciplines has led to widespread adoption of machine learning (ML) for prediction. However, most ML algorithms focus on point estimation and provide limited information about predictive uncertainty or the conditional distribution of the response, restricting their ability to characterize rare or extreme outcomes. We develop QXGB, a quantile-based gradient boosting framework, and introduce a convolution smoothed loss within it that estimates conditional quantiles for constructing dense cumulative distribution functions (CDFs), exceedance probabilities, and tail behaviour relevant to extreme outcomes. This approach preserves the computational efficiency of extreme gradient boosting while restoring the Hessian information XGBoost relies on for tree splitting, in turn providing interpretable measures of extreme value and exceedance probability predictions. We derive the gradients and Hessians needed to integrate convolution smoothed quantile loss with different kernel specifications into XGBoost, and with simulated data, benchmark this approach against alternative smoothed quantile regression losses, the native quantile objective in the XGBoost Python package, and independent versus multi-output tree estimation. The practical relevance is illustrated in an application predicting fine particulate matter (PM $_{2.5}$ ) in northern California, including periods where levels were elevated due to wildfire smoke. Our results show that convolution smoothed QXGB, particularly when paired with multi-output trees, delivers accurate predictions with near-zero quantile crossing, well-calibrated CDF and exceedance probability estimates, and useful tail characterization for extreme values. Interval estimation is also evaluated as a measure of data spread.

Figures & tables

Figure 1: Simulated daily PM 2.5 concentrations ( $\mu$ g/m 3 ) for one year, with more extreme levels during the wildfire season.

Method	Smoothing Parameter	Average CRPS	Average Crossing Rate
Default Python	0.1	2.72	0.12
Huber Pinball		2.77	0.15
Arctan Pinball		2.74	0.01
Convolution (Laplace)		2.74	0.02
Convolution (Gaussian)		2.75	0.03
Default Python	0.5	2.72	0.12

Table 1: Average CRPS and Crossing Rates across all 50 simulated locations.

Model	Tuned Smoothing Parameter	$\bm{N_{obs}}$	$\bm{N_{pred}}$	BSS	ROC-AUC
Logistic (Binary Baseline)		845	3962.50	-0.87	0.80
Method 1: QXGB (Default)		845	919.00	0.30	0.82
Method 2: QXGB (Huber)	t=0.1	845	806.50	0.30	0.81
Method 3: QXGB (Arctan)	s=0.05	845	1060.6	0.30	0.82
Method 4: QXGB (Gaussian Tuned)	h=0.2	845	890.30	0.31	0.81
Method 5: QXGB (Laplace Tuned)	h=0.1	845	850.80	0.31	0.81

Table 2: Exceedance Probability Results.

Simulated Data	Python	Huber Pinball	Arctan Pinball	Convolution (Laplace)	Convolution (Gaussian)
With Spike
PICP	0.86	0.73	0.97	0.91	0.94
PINAW	0.18	0.14	0.25	0.19	0.20
Interval Score	49.48	55.71	53.86	50.26	51.20
Tuned Smoothing Parameter		t=0.5	s=0.1	h=0.1	h=0.25

Table 3: PICP, PINAW, and interval score for 90% intervals formed from the 0.05 and 0.95 quantiles

Panel of Figure 2 — Figure 2: Description of the PM 2.5 data (a) Locations of the 53 sites in Northern California used in this study, colored dots match (b), time series of the daily concentrations for the 10 monitoring sites with highest levels across 2012-2018.

Method	Smoothing Parameter	Average CRPS	Average Crossing Rate
Default Python	0.1	1.18	0.22
Huber Pinball		2.65	0.75
Arctan Pinball		1.20	0.02
Convolution (Laplace)		2.03	0.45
Convolution (Gaussian)		2.53	0.57
Default Python	0.5	1.18	0.22

Table 4: Dense CDF evaluation on the real PM 2.5 dataset using spatial LOSO-CV.

Model	Tuned Smoothing Parameter	$\bm{N_{obs}}$	$\bm{N_{pred}}$	BSS	ROC-AUC
Logistic (Binary Baseline)		736	1343.90	0.41	0.99
Method 1: QXGB (Default)		736	672.40	0.66	0.98
Method 2: QXGB (Huber)	t=1.0	736	643.40	0.69	0.98
Method 3: QXGB (Arctan)	s=0.1	736	977.40	0.67	0.98
Method 4: QXGB (Gaussian Tuned)	h=0.5	736	747.20	0.70	0.99
Method 5: QXGB (Laplace Tuned)	h=0.2	736	687.90	0.69	0.99

Table 5: Exceedance Probability Results.

Metric	Python	Huber Pinball	Arctan Pinball	Convolution (Laplace)	Convolution (Gaussian)
PICP	0.87	0.74	1.00	0.97	0.98
PINAW	0.14	0.10	1.02	0.24	0.31
Interval Score	20.61	24.40	84.69	25.55	29.97
Tuned Smoothing Parameter		t=3.0	s=0.1	h=0.25	h=0.5

Table 6: PICP and PINAW metrics for the intervals made by estimating the 0.05 and 0.95 quantiles using the 5 methods mentioned previously. This was done for the data from 2018.

Panel of Figure 3 — Figure 3: Plots of 2018 PM 2.5 levels in Northern California, with lines showing the estimates of the (a) lower bounds of the intervals generated by the 5 approaches and (b) upper bounds of the intervals generated by the 5 approaches.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Smoothing Parameter	PICP	PINAW	Interval Score
Huber	$0.1$	$0.82$	$0.18$	$57.75$
	$0.25$	$0.77$	$0.16$	$52.85$
	$0.5$	$0.73$	$0.14$	$55.71$
	$0.75$	$0.70$	$0.14$	$56.81$
	$1.0$	$0.69$	$0.13$	$57.69$
Arctan pinball loss	$0.1$	$0.97$	$0.25$	$53.86$

Appendix

Table A1 : Combined performance metrics with varying smoothing parameters and intervals

Model	Tuned Smoothing Parameter	$\bm{N_{obs}}$	$\bm{N_{pred}}$	BSS	ROC-AUC
Logistic (Binary Baseline)		386	2114.90	-0.84	0.81
Method 1: QXGB (Default)		386	1067.70	0.25	0.75
Method 2: QXGB (Huber)	t=0.5	386	1064.80	0.24	0.73
Method 3: QXGB (Arctan)	s=0.2	386	1114.90	0.24	0.79
Method 4: QXGB (Gaussian Tuned)	h=0.05	386	1059.80	0.24	0.74
Method 5: QXGB (Laplace Tuned)	h=0.5	386	1089.60	0.25	0.76

Appendix

Table A2 : Exceedance Probability Results for Threshold=50.0

Simulated Data	Python	Huber Pinball	Arctan Pinball	Convolution (Laplace)	Convolution (Gaussian)
With Spike
PICP	0.86	0.75	0.96	0.95	0.93
PINAW	0.19	0.15	0.24	0.22	0.20
Interval Score	50.42	57.48	53.36	52.80	51.02
Tuned Smoothing Parameter		t=0.25	s=0.1	h=0.25	h=0.25

Appendix

Table A3 : PICP, PINAW, and interval score for 90% prediction intervals formed from the 0.05 and 0.95 quantiles, quantiles computed individually

Method	Smoothing Parameter	Average CRPS	Average Crossing Rate
Default Python	0.1	2.71	0.13
Huber Pinball		2.79	0.69
Arctan Pinball		2.74	0.19
Convolution (Laplace)		2.74	0.36
Convolution (Gaussian)		2.75	0.43
Default Python	0.5	2.71	0.13

Appendix

Table A4 : Average CRPS and Crossing Rates across all 50 simulated locations, quantiles computed individually

Metric	Python	Huber Pinball	Arctan Pinball	Convolution (Laplace)	Convolution (Gaussian)
PICP	0.91	0.74	0.99	0.97	0.98
PINAW	0.26	0.11	0.64	0.24	0.30
Interval Score	32.56	25.89	66.11	25.39	30.00
Tuned Smoothing Parameter		t=10.0	s=0.1	h=0.25	h=0.5

Appendix

Table A5 : PICP and PINAW metrics for the prediction intervals made by estimating the 0.05 and 0.95 quantiles using the 5 methods mentioned previously. This was done for the data from 2018, quantiles computed individually

Method	Smoothing Parameter	Average CRPS	Average Crossing Rate
Default Python	0.1	1.33	0.36
Huber Pinball		2.60	0.97
Arctan Pinball		1.20	0.29
Convolution (Laplace)		1.82	0.75
Convolution (Gaussian)		2.32	0.84
Default Python	0.5	1.33	0.36

Appendix

Table A6 : Dense CDF evaluation on the real PM 2.5 dataset using spatial LOSO-CV, quantiles computed individually

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